Diffusion Equations on Polyhedral Meshes with Mixed Cells

Diffusion Equations on Polyhedral Meshes with Mixed Cells Yuri Kuznetsov Department of Mathematics University of Houston http: //lacsi. rice. edu/review/slides_2006 LACSI Review, February 2006

LACSI Project: 2005 Advanced Numerical Methods for Diffusion Equations in Heterogeneous Media on Distorted Polyhedral Meshes LANL: M. Shashkov – PI K. Lipnikov, D. Moulton, S. Runnels UH: Y. Kuznetsov – PI O. Boyarkin – Post. Doc V. Gvozdev, D. Svyatskiy – Graduate Students S. Repin – Visiting Research Professor

OUTLINE • Problem Formulation • Meshes with Mixed Cells • New Polyhedral Discretization • Numerical Results • Applications to Homogenization and AMR • Project Activities • Further Research Plans

Diffusion Equation Here, Ω -- polyhedral computational domain K -- diffusion tensor c -- nonnegative coefficient n -- outward normal F -- source function

First Order System • Flux Equation ( Darcy Law ) or • Conservation Law Equation

Polyhedral H-mesh where {Ei} – polyhedral cells Polyhedral cell E

Mixed Cells

Discrete Conservation Law (1 a) By integration over a polyhedral cell E: we get the discrete equation where

Discrete Conservation Law (1 b)

Discrete Conservation Law (2) where *) *) Ref. to M. Shashkov for Mimetic Finite Difference methods

Major Target (1) To design a discretization for the diffusion equation with discontinuous K, c, and F in a polyhedral cell E under the following conditions: -- one DOF per Γk for the normal flux; -- one DOF per cell E for the solution function.

Major Target (2) To design a discretization for the flux equation in the form where and GH is an explicitly computed mesh function.

Polyhedral Discretizations 2003/2004 Major assumptions: Major advantages: • Arbitrary diffusion tensor • Arbitrary polyhedral meshes including meshes with nonconvex and degenerated cells • Nonmatching and AMR polyhedral meshes

Impact on ASC Projects LANL researchers M. Shashkov and K. Lipnikov in T 7 group, and S. Runnels in X 3 group have recently implemented the proposed polyhedral discretization scheme for the diffusion equations in FLAG code for SHAVANO project. A parallel version of the code was developed by K. Lipnikov and S. Runnels in cooperation with other members of X 3 group.

New Polyhedral Discretization on Mixed Cells (1) Consider the diffusion equation on a polyhedral H-cell E: with the boundary conditions Polyhedral h-partitioning of E where {ej} are polyhedral h-cells.

Mixed Cells

Triangulated Mixed Cells

New Polyhedral Discretization on Mixed Cells (2) Discretization in Eh where (*) -- flux equations on the boundary ΓH of E (**) -- flux equations in the interior of E (***) -- conservation law equations in Eh

New Polyhedral Discretization on Mixed Cells (3) The mesh operator is nonsingular. Thus,

New Polyhedral Discretization on Mixed Cells (4) Substituting uh and ph in the flux equation on ΓH we get the equation where and

New Polyhedral Discretization on Mixed Cells (5) Major Theoretical Result where Reminder: The discrete conservation law in E

New Polyhedral Discretization on Mixed Cells (6) Hybrid system in terms of u. H, p. H, and pΓ, H + interface conditions for the normal fluxes Assembled system in terms of u. H and p. H

Test 1

Error for Solution Function New Scheme Old Scheme 16 x 16 0. 0011 0. 026 32 x 32 0. 0003 0. 0038 64 x 64 0. 000078 0. 0012 128 x 128 0. 000019 0. 00036

Error for Flux New Scheme Old Scheme 16 x 16 0. 76 9. 40 32 x 32 0. 38 5. 57 64 x 64 0. 19 4. 11 128 x 128 0. 096 3. 35

Test 2

Error for Solution Function

Error for Flux New Scheme Old Scheme 16 x 16 2. 6 16. 31 32 x 32 1. 35 8. 49 64 x 64 0. 6 4. 32 128 x 128 0. 29 3. 1

Heterogeneous H-cell

Multilevel Homogenization Original H-cell Refined mesh, H/4 Refined mesh, H/2 Refined mesh, H/8

Multilevel AMR Mesh

LANL -- UH Communication • LACSI Symposium, Santa Fe 2005: Workshop on advanced numerical methods for PDEs LANL presentations: K. Lipnikov, T 7 D. Moulton, T 7 S. Runnels, X 3 M. Shashkov, T 7 J. Warsa, CCS 4 UH presentations: V. Gvozdev – Ph. D. student Y. Kuznetsov D. Svyatskiy – Ph. D. student • Meeting at UH, January 11— 14, 2006 Attendees: LANL: M. Shashkov UH: Y. Kuznetsov, O. Boyarkin, D. Svyatskiy

Education Issues (1) Konstantin Lipnikov: 2001 & 2002: -- summer semesters at LANL (Ph. D. Thesis – 2002) 2002 -- 2004: -- Post. Doc at T 7 group, LANL since January 2005: -- limited term staff member at T 7 group, LANL Vadim Dyadechko: 2002 & 2003: -- summer semesters at LANL (Ph. D. Thesis – 2003) since September 2003: -- Post. Doc at T 7 group, LANL

Education Issues (2) Oleg Boyarkin: 2001— 2004: -- Graduate Student at UH supported by LACSI 2004: -- Ph. D. Thesis since January 2005: -- Post. Doc at Department of Mathematics, UH Daniil Svyatskiy: 2004 & 2005: -- summer semesters at LANL Plan: -- Ph. D. Thesis – April 2006 -- Post. Doc at T 7 group, LANL – from June 2006

Further Research Plans • 3 D evaluation of new polyhedral discretizations • Applications to AMR • Multilevel preconditioners based on polyhedral discretizations • Discretizations on anisotropic polyhedral meshes